(1) common factor: the common factor of each term is called ~ of this polynomial term.
② Extraction method of common factor: Generally speaking, if every term of a polynomial has a common factor, you can put this common factor outside brackets and write the polynomial as a factor product. This method of decomposing factors is called extracting common factors.
am+bm+cm=m(a+b+c)
③ Specific methods: When all the coefficients are integers, the coefficients of the common factor formula should take the greatest common divisor of all the coefficients; The letter takes the same letter for each item, and the index of each letter takes the lowest degree. If the first term of a polynomial is negative, a "-"sign is usually put forward to make the coefficient of the first term in brackets positive.
⑵ Use the formula method.
① variance formula:. a 2-b 2 = (a+b) (a-b)
② Complete square formula: a 2 2ab+b 2 = (a b) 2.
Polynomials that can be decomposed by the complete square formula must be trinomials, two of which can be written as the sum of squares of two numbers (or formulas), and the other is twice the product of these two numbers (or formulas). ※ 。
③ Cubic sum formula: A 3+B 3 = (A+B) (A 2-AB+B 2).
Cubic difference formula: a 3-b 3 = (a-b) (a 2+ab+b 2).
④ Complete cubic formula: a 3 3a 2b+3ab 2 b 3 = (a b) 3.
⑤a^n-b^n=(a-b)[a^(n- 1)+a^(n-2)b+……+b^(n-2)a+b^(n- 1)]
A m+b m = (a+b) [a (m-1)-a (m-2) b+...-b (m-2) a+b (m-1)] (m is an odd number).
⑶ Grouping decomposition method
Grouping decomposition: a method of grouping polynomials and then decomposing factors.
The grouping decomposition method must have a clear purpose, that is, the common factor can be directly extracted or the formula can be used after grouping.
(4) Methods of splitting and supplementing projects
Decomposition and supplement method: one term of polynomial is decomposed or filled with two terms (or several terms) which are opposite to each other, so that the original formula is applicable to common factor method, formula method or group decomposition method; It should be noted that the deformation must be carried out under the principle of equality with the original polynomial.
5] Cross multiplication.
① factorization of x2+(p q) x+pq formula.
The characteristics of this kind of quadratic trinomial formula are: the coefficient of quadratic term is1; Constant term is the product of two numbers; The coefficient of a linear term is the sum of two factors of a constant term. So we can directly decompose some quadratic trinomial factors with the coefficient of 1: x 2+(p q) x+PQ = (x+p) (x+q).
② Factorization of KX2+MX+N formula
If it can be decomposed into k = AC, n = BD and AD+BC = M, then
kx^2+mx+n=(ax b)(cx d)
a \ - /b ac=k bd=n
c / - \d ad+bc=m
General steps of polynomial decomposition. ※:
(1) If the polynomial term has a common factor, then the common factor should be raised first;
(2) If there is no common factor, try to decompose it by formula and cross multiplication;
(3) If the above methods cannot be decomposed, you can try to decompose by grouping, splitting and adding items;
④ Factorization must be carried out until each polynomial factor can no longer be decomposed.
(6) Applying factorial theorem: If f(a)=0, then f(x) must contain factorial (x-a). If f (x) = x 2+5x+6 and f(-2)=0, it can be determined that (x+2) is a factor of x 2+5x+6.